Analytic and Geometric Theories of Holomorphic and CR Mappings John Bland (University of Toronto), Herve Gaussier (Universite´ de Provence), Kang-Tae Kim (Pohang University of Science and Technology), Steven G. Krantz (Washington University), Finnur Larusson (University of Western Ontario), Junjiro Noguchi (University of Tokyo) April 29, 2006–May 4, 2006 1 Overview of the Field Several complex variables is a modern and dynamic part of mathematics. Spawned in the early twentieth century by William Fogg Osgood (United States), Henri Poincare´ (France), Fritz Hartogs (Germany), and others, the subject has always had an international flavor. Certainly our workshop reflected that flavor, as we had participants from Korea, Japan, Australia, Russia, Iceland, the U.S.A., France, and India. All of the participants already knew each other—at least professionally—but it was a special pleasure to meet face-to- face and to exchange ideas spontaneously and in real time. Many of the participants are young mathematicians on the cutting edge of current research. It is especially important for a broad cross-section of workers in the field to be able to interact with these important researchers. That is what our workshop achieved. The two seminal results proved at the inception of this subject are these: Theorem (Poincare):´ Let 2 2 2 B = {(z1, z2) ∈ C : |z1| + |z2| < 1} be the unit ball in C2 and let 2 2 D = {(z1, z2) ∈ C : |z1| < 1, |z2| < 1} be the bidisc. Then there is no biholomorphic mapping Φ : B → D2 . This theorem tells us that any obvious guess at a Riemann mapping theorem in several complex variables will fail—just because the two most obvious candidates for the “default” domain are not equivalent. Theorem (Hartogs): Let 2 2 Ω = {(z1, z2) ∈ C : |z1| < 2, |z2| < 2} \ {(z1, z2) ∈ C : |z1| ≤ 1, |z2| ≤ 1} . Let f be a holomorphic function on Ω. Then there exists a holomorphic function F on 2 D = {(z1, z2) : |z1| < 2, |z2| < 2} 1 2 such that . F Ω = f The result of Hartogs says that Ω is not the natural domain of definition of any holomorphic function; for any holomorphic function on Ω analytically continues to a strictly larger domain. This theorem should be contrasted with the situation in C1: For any domain Ω ⊆ C1, there is a holomorphic function f on Ω that cannot be analytically continued to a larger domain. A domain in any Cn which is the natural domain of definition of some holomorphic function is called a domain of holomorphy. Thus the focus of several complex variables for the past century has been on holomorphic mappings, and on domains of holomorphy. A vast array of techniques has been developed for coming to grips with these two circles of ideas. Today several complex variables interacts profitably with harmonic analysis, partial differential equations, differential geometry, commutative algebra, algebraic geometry, one complex variable, real variable theory, and even formal logic. Among the methodologies that have been developed to tackle the questions that have been described are (i) Sheaf theory (J. Leray and K. Oka); (ii) Partial differential equations, notably the ∂ problem (J. J. Kohn [13] and L. Hormander¨ [12]); (iii) Methods of differentiable geometry, notably Kahler¨ manifold theory and Stein manifold theory (S. S. Chern, H. Grauert, R. Narasimhan, H. Remmert). These powerful pieces of mathematical machinery have proved to be useful in a variety of mathematical contexts. They continue to be developed today. Furthermore, more recondite techniques such as the Monge- Ampere` equation, dynamical systems, Banach algebras, function algebras, operator techniques, and many other widespread ideas continue to develop alongside the principal streams of thought. All these branches of the subject interact profitably, and produce a colorful and productive melange of mathematical work. 2 Recent Developments and Open Problems In the past twenty years, the subject of several complex variables has blossomed in a variety of new directions. Among these are • The theory of dynamical systems in several complex variables. Of course dynamical systems found their genesis in the work of H. Poincare´ on questions of celestial mechanics. But in fact it was quite early in the twentieth century that Fatou and Julia saw the relevance of these new ideas to complex function theory (in one complex variable). In the hands of Hubbard, Douady, Mandelbrot, and many others, one-variable dynamical systems has blossomed into a vigorous part of modern mathematics. The growth of dynamical systems in the several-variable setting is considerably newer. And the questions are quite a lot harder. Deep ideas from pluripotential theory, differential geometry, partial differential equations, and many other parts of mathematics must be brought to bear in order to achieve any progress at all. Major workers in the area include Bedford (present at the workshop), Buzzard, Fornaess, Sibony, Lyubich, and Douady. • The theory of automorphism groups of domains. It has already been noted that, because of Poincare’s´ theorem, we can expect no version of the Riemann mapping theorem in several complex variables. In fact Poincare’s´ stunning result has been enhanced and refined in the ensuing years. Work of Chern- Moser [7], Burns-Shnider-Wells [6], and Greene-Krantz (organizer of this conference) [10], [11] has shown that the phenomenon that Poincare´ discovered is in fact generic in a variety of senses. But Poincare´ himself gave us some guidance as to how to come to grips with the phenomenon. He proved his result by analyzing the automorphism groups of the two domains D2 and B. Here, if Ω ⊆ Cn is a domain, then the automorphism group Aut(Ω) is the collection of biholomorphic self- maps of Ω. This set forms a group under the binary operation of composition of mappings. And in fact, if we equip the automorphism group with the topology of uniform convergence on compact sets (equivalently, the compact-open topology), then (at least for a bounded domain Ω) the automorphism 3 groups turns out to be a real Lie group. Thus powerful bodies of machinery may be brought to bear on the study of automorphism groups. For a given domain Ω, one may study algebraic, topological, and analytic properties of Aut(Ω) and determine how they reflect the complex geometry of Ω (and vice versa). This study, which has been particularly vigorous in the past twenty-five years, has become an ersatz for the Riemann mapping theorem: it gives us a device for differentiating and comparing domains. Major workers in the field include Bedford (present at the workshop), Kim (present at the workshop, and an organizer), Krantz (present at the workshop, and an organizer), Pinchuk, Greene, Berteloot (present at the workshop), and Kodama (present at the workshop). • CR geometry, analysis of hypersurfaces, and normal forms. Poincare´ advocated a program of studying the equivalence and inequivalence of domains in Cn by (i) first proving that any biholomor- phic mapping of domains will extend smoothly to the boundaries and (ii) then constructing differential invariants on the boundaries. It turned out that the mathematical techniques were not yet available to carry out step (i), and it awaited Charles Fefferman to (in 1974 [8]) prove that a biholomorphic map- ping of strongly pseudoconvex domains continues smoothly to the boundaries. His work was quickly followed by work of Chern and Moser on constructing the anticipated differential boundary invariants. Fefferman later used these invariants to develop a classification theory for strongly pseudoconvex do- mains [9]. Bell and Ligocka [], [] were able to extend and simplify Fefferman’s theorem, and to turn it into a powerful and versatile tool for our subject. Today the study of biholomorphic mappings, and the cognate idea known as Condition R, is a vital part of several complex variables. Work continues on developing these ideas for broader classes of domains. The methods involve com- mutative algebra, differential geometry, and partial differential equations. Principal workers in the field today include Gong, Hayashimoto (present at the workshop), Isaev (present at the workshop), Ezhov, and Schmalz (present at the workshop). • Holomorphic functions and mappings in infinite dimensions. The idea of studying holomorphic functions and mappings on Banach spaces is more than fifty years old. Certainly the functional calculus and other natural questions of operator theory begged for such a development. But it must be said that little substantial progress was made in the subject until relatively recently. The difficulty, it seems, is that people were trying to study holomorphic functions and mappings on any Banach space. What we have learned in the past ten years—thanks to work of Lempert and others—is that considerable progress can be made if we restrict attention to particular Banach spaces. There are now a theory of domains of holomorphic, a theory of normal families, a theory of the ∂ problem, and many other important cornerstones of function theory in the infinite-dimensional setting. Among the experts in this subject area are Cima, Graham (present at the workshop), Kim (present at the workshop), Krantz (present at the workshop), and Lempert. • CR functions and mappings In the 1960s, Kohn and Rossi introduced the concept of a CR function. This is a function on a real hypersurface in Cn that is in effect the “trace” of a holomorphic function. One desires a definition of such an object that is intrinsic, and makes no reference to holomorphic functions; this is achieved by way of partial differential equations. Thus are defined the CR functions. Also CR mappings are defined similarly. In the intervening forty years, CR functions and mappings have assumed a prominent role in the subject. They are useful for studying the ∂ and ∂b problems, for studying holomorphic mappings, and for developing function theory. Certainly this subject area was one of the focuses of our conference.